The proofs of Lowenheim-Skolem I have seen all depended on the use of choice functions. Is there any proof not dependent on axiom of choice? Or is Lowenheim-Skolem a result of axiom of choice?
2026-04-03 06:06:42.1775196402
Does Lowenheim-Skolem theorem depend on axiom of choice?
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The Skolem-Lowenheim theorem is equivalent to $\sf DC$, which is a strong countable choice principle.
More generally, the following is true:
You can find the proof here. In a nutshell, the idea is that given a family of sets, an elementary substructure which is well-orderable gives you a way to define a choice function. The other direction is the usual proof of Skolem-Lowenheim.